OTC 2L ns

Omnitetrachordal scales of the form 2L+ns -- a special case?

An OTC scale of this form (example: LssssLsssssss) has been confirmed for every scale size from 3 to 53 (except 4), and probably exists for any larger scale size as well. With the exception of the 3-, 5-, and 7-note scales, these patterns are not MOS.

In every case so far studied, the strings of 's' steps come in a small group and a large group. Let x be the number of 's' steps in the small group, and y the number of 's' steps in the large group. Then:

• 9/8 = (y-x)s
• 4/3 = L+xs
• 3/2 = L+ys
• If either x or y is odd, the scale has a symmetric mode (ex. sLsLs).
• If both x and y are odd, there are two symmetric modes (ex. sLssssLs and ssLssLss).
• If both x and y are even, no symmetric mode exists (ex. LsLsss).

As scale size increases:

• y/x appears to converge on a value around 0.6 .
• For odd size scales, the generator is large (at least 4/3 = 498 cents), and appears to approach sqrt(2) = 600 cents.
• For even size scales, the period is 1/2 octave, and the generator is small, and appears to approach 1/1 = 0 cents.

Note the patterns in values of P:

• P(3) = P(5)+1 = P(7)+2
• P(6) = P(8)+1 = P(10)+2 = P(12a)+3
• P(9) = P(11)+1 = P(13)+2 = P(15)+3 = P(17)+4
• P(12b) = P(14)+1 = P(16)+2 = P(18)+3 = P(20)+4 = P(22)+5
• P(19) = P(21)+1 = P(23)+2 = P(25)+3 = P(27)+4 = P(29)+5
• etc.

These groups appear to share generators and temperaments.

For some scale sizes (12, 17, 22, 29, 34, 41, 46, 53, and possibly others), more than one class of OTC scale is possible! In such cases, the two will have different tetrachordal divisions (i.e. representing 4/3 with different numbers of steps). One version will have P very near 1, and the other will have a much higher value of P. Note that these scale sizes tend to correspond to EDOs that represent 4/3 and 3/2 with relatively high accuracy.

size	2L+_s	P	pattern
3	2L+s	2.4424745962	L s L (MOS)
5	2L+3s	1.4424745962	Ls s Ls (MOS)
6	2L+4s	3.8849491924	Ls ss Ls
7	2L+5s	0.4424745962	Lss s Lss (MOS)
8	2L+6s	2.8849491924	Lss ss Lss
9	2L+7s	5.3274237885	Lss sss Lss
10	2L+8s	1.8849491924	Lsss ss Lsss
11	2L+9s	4.3274237885	Lsss sss Lsss
12a	2L+10s	0.8849491924	Lssss ss Lssss
12b	2L+10s	6.7698983847	Lsss ssss Lsss
13	2L+11s	3.3274237885	Lssss sss Lssss
14	2L+12s	5.7698983847	Lssss ssss Lssss
15	2L+13s	2.3274237885	Lsssss sss Lsssss
16	2L+14s	4.7698983847	Lsssss ssss Lsssss
17a	2L+15s	1.3274237885	Lssssss sss Lssssss
17b	2L+15s	7.2123729809	Lsssss sssss Lsssss
18	2L+16s	3.7698983847	Lssssss ssss Lssssss
19	2L+17s	6.2123729809	Lssssss sssss Lssssss
20	2L+18s	2.7698983847	Lsssssss ssss Lsssssss
21	2L+19s	5.2123729809	Lsssssss sssss Lsssssss
22a	2L+20s	1.7698983847	Lssssssss ssss Lssssssss
22b	2L+20s	7.6548475771	Lsssssss ssssss Lsssssss
23	2L+21s	4.2123729809	Lssssssss sssss Lssssssss
24	2L+22s	6.6548475771	Lssssssss ssssss Lssssssss
25	2L+23s	3.2123729809	Lsssssssss sssss Lsssssssss
26	2L+24s	5.6548475771	Lsssssssss ssssss Lsssssssss
27	2L+25s	2.2123729809	Lssssssssss sssss Lssssssssss
28	2L+26s	4.6548475771	Lssssssssss ssssss Lssssssssss
29a	2L+27s	1.2123729809	Lsssssssssss sssss Lsssssssssss
29b	2L+27s	7.0973221733	Lssssssssss sssssss Lssssssssss
30	2L+28s	3.6548475771	Lsssssssssss ssssss Lsssssssssss
31	2L+29s	6.0973221733	Lsssssssssss sssssss Lsssssssssss
32	2L+30s	2.6548475771	Lssssssssssss ssssss Lssssssssssss
33	2L+31s	5.0973221733	Lssssssssssss sssssss Lssssssssssss
34a	2L+32s	1.6548475771	Lsssssssssssss ssssss Lsssssssssssss
34b	2L+32s	6.5397967694	Lssssssssssss ssssssss Lssssssssssss
35	2L+33s	4.0973221733	Lsssssssssssss sssssss Lsssssssssssss
36	2L+34s	6.5397967694	Lsssssssssssss ssssssss Lsssssssssssss
37	2L+35s	3.0973221733	Lssssssssssssss sssssss Lssssssssssssss
38	2L+36s	5.5397967694	Lssssssssssssss ssssssss Lssssssssssssss
39	2L+37s	2.0973221733	Lsssssssssssssss sssssss Lsssssssssssssss
40	2L+38s	4.5397967694	Lsssssssssssssss ssssssss Lsssssssssssssss
41a	2L+39s	1.0973221733	Lssssssssssssssss sssssss Lssssssssssssssss
41b	2L+39s	6.9822713656	Lsssssssssssssss sssssssss Lsssssssssssssss
42	2L+40s	3.5397967694	Lssssssssssssssss ssssssss Lssssssssssssssss
43	2L+41s	5.9822713656	Lssssssssssssssss sssssssss Lssssssssssssssss
44	2L+42s	2.5397967694	Lsssssssssssssssss ssssssss Lsssssssssssssssss
45	2L+43s	4.9822713656	Lsssssssssssssssss sssssssss Lsssssssssssssssss
46a	2L+44s	1.5397967694	Lssssssssssssssssss ssssssss Lssssssssssssssssss
46b	2L+44s	7.4247459618	Lsssssssssssssssss ssssssssss Lsssssssssssssssss
47	2L+45s	3.9822713656	Lssssssssssssssssss sssssssss Lssssssssssssssssss
48	2L+46s	6.4247459618	Lssssssssssssssssss ssssssssss Lssssssssssssssssss
49	2L+47s	2.9822713656	Lsssssssssssssssssss sssssssss Lsssssssssssssssssss
50	2L+48s	5.4247459618	Lsssssssssssssssssss ssssssssss Lsssssssssssssssssss
51	2L+49s	1.9822713656	Lssssssssssssssssssss sssssssss Lssssssssssssssssssss
52	2L+50s	4.4247459618	Lssssssssssssssssssss ssssssssss Lssssssssssssssssssss
53a	2L+51s	0.9822713656	Lsssssssssssssssssssss sssssssss Lsssssssssssssssssssss
53b	2L+51s	6.8672205580	Lssssssssssssssssssss sssssssssss Lssssssssssssssssssss

etc.

See also

• Omnitetrachordality
• Gallery of omnitetrachordal scales